A038338 → x ↦ x+1 → (θ+1)²

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Submission text ?

Triangle begins:
  0:              576,              2304,               1296;
  1:            13824,             82944,              93312,              27648;
  2:           331776,           2654208,            4478976,            2654208,             518400;
  3:          7962624,          79626240,          179159040,          159252480,           62208000,            8957952;
  4:        191102976,        2293235712,         6449725440,         7644119040,         4478976000,         1289945088,          146313216;
  5:       4586471424,       64210599936,       216710774784,       321052999680,       250822656000,       108355387392,        24580620288,         2293235712;
  6:     110075314176,     1761205026816,      6934744793088,     12328435187712,     12039487488000,      6934744793088,      2359739547648,       440301256704,       34828517376;
  7:    2641807540224,    47552535724032,    213986410758144,    443823666757632,    520105859481600,    374476218826752,    169901247430656,     47552535724032,     7522959753216,      515978035200;
  8:   63403380965376,  1268067619307520,   6419592322744320,  15216811431690240,  20804234379264000,  17974858503684096,  10194074845839360,   3804202857922560,   902755170385920,   123834728448000,    7492001071104;
  9: 1521681143169024, 33476985149718528, 188308041467166720, 502154777245777920, 784616839446528000, 790893774162100224, 538247151860318208, 251077388622888960, 79442454993960960, 16346184155136000, 1977888282771456, 106993205379072;

The row polynomials satisfy (4+(6*n+10)*x+(n^2+4*n+4)*x^2)*P(n,x) = (96+(144*n+432)*x+(24*n^2+216*n+408)*x^2+(12*n^2+72*n+108)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 576 + 2304*x + 1296*x^2.

P[0, x_] := 576 + 2304*x + 1296*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((96 + (144*n + 432)*x + (24*n^2 + 216*n + 408)*x^2 + (12*n^2 + 72*n + 108)*x^3)*P[n - 1, x])/(4 + (6*n + 10)*x + (n^2 + 4*n + 4)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 576 + 2304*x + 1296*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((96 + (144*n + 432)*x + (24*n^2 + 216*n + 408)*x^2 + (12*n^2 + 72*n + 108)*x^3)*v[i - 1])/(4 + (6*n + 10)*x + (n^2 + 4*n + 4)*x^2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

Properties

• Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
  by Laguerre’s theorem on multiplier sequences
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on multiplier sequences
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Laguerre’s theorem on multiplier sequences
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
  by Laguerre’s theorem on multiplier sequences

Proof chains ? JSON

• Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›

  • from basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*12^j.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*12^j.?
        • Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check
        • Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
        • Parent-row provers of polytool polytool ? step › direct check
        • Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*12^j.?
            • ↑ Row-hypergeometric rows (multiplier sequences) (above)
            • ↑ Liu–Wang theorem (above)
            • ↑ Parent-row provers of polytool (above)
            • ↑ Iterated real-rootedness preservers (above)
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*12^j.?
        • ↑ Row-hypergeometric rows (multiplier sequences) (above)
        • ↑ Liu–Wang theorem (above)
        • ↑ Parent-row provers of polytool (above)
        • ↑ Iterated real-rootedness preservers (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)

Related ?

Same fingerprint ?

• A178820 → basis A007318 → (θ+1)²
• A028326 → basis A007318 → (θ+1)²
• A120910 → (θ+1)²
• A038325 → basis A007318 → (θ+1)²
• A038330 → basis A013609 → (θ+1)²
• A356546 → x ↦ x+1 → (θ+1)²
• A178822 → x ↦ x+1 → (θ+1)²
• A135065 → x ↦ x+1 → (θ+1)²
• A038207 → (θ+1)²
• A085880 → x ↦ x+1 → (θ+1)²
• A027466 → basis A013611 → (θ+1)²
• A156992 → basis A007318 → (θ+1)²
• A178821 → basis A007318 → (θ+1)²
• A038221 → x ↦ x+1 → (θ+1)²
• A120910 → D → (θ+1)²
• basis-map coefficients of A113214, shifted → x ↦ x+1 → (θ+1)²
• A003506 → x ↦ x+1 → (θ+1)²
• A196347 → x ↦ x+1 → (θ+1)²
• A085881 → basis A007318 → (θ+1)²
• A038208 → basis A007318 → (θ+1)²

References

1. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
3. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.